Identification of Green's Functions Singularities by Cross Correlation of Noisy Signals


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Abstract: In this paper we consider the problem of estimating the singular support of the Green's function of the wave equation in a bounded region by cross correlating noisy signals. A collection of sources with unknown spatial distribution emit stationary random signals into the medium, which are recorded at two observation points. We show that the cross correlation of these signals has enough information to identify the singular component of the Green's function, which provides an estimate of the travel time between the two observation points. As in the recent work of Y. Colin de Verdière [math-ph/0610043], we use semiclassical arguments to approximate the wave dynamics by classical dynamics. Next we use the ergodicity of the ray dynamics to obtain an explicit expression of the cross correlation of the noisy signals. We also show that this approach is statistically stable when the averaging time is long enough, and that the accuracy of the travel time estimation is directly related to the spatial correlation function of the sources.

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Date: 2007-11-14